Search NASA⌕ Search

SEARCH · Search NASA

Results for “analyzable function”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 631 records · Page 35
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Electron scattering on ${\mathbf{A=3}}$ nuclei from quantum Monte Carlo based approaches

We perform first-principle calculations of electron-nucleus scattering on $^3$He and $^3$H using the Green's function Monte Carlo method and two approaches based on the factorization of the final hadronic state: the spectral-function formalism and the short-time approximation. These three methods are benchmarked among each other and compared to the experimental data for the longitudinal and transverse electromagnetic response functions of $^3$He, and the inclusive cross sections of both $^3$He and $^3$H. Since these three approaches are based on the same description of nuclear dynamics of the initial target state, comparing their results enables a precise quantification of the uncertainties inherent to factorization schemes. At sufficiently large values of the momentum transfer, we find an excellent agreement of the Green's function Monte Carlo calculation with experimental data and with both the spectral-function formalism and the short-time approximation. We also analyze the relevance of relativistic effects, whose inclusion becomes crucial to explain data at high momentum and energy transfer.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2018

Rn gravity as a viable alternative to dark matter: Application to stellar dynamics

The role of f(R) gravity, as well as the other modifications of standard Einstein’s gravity, is to explain the accelerated expansion, structure formation of the Universe, and some other phenomena at extragalactic scales (such as e.g. flat rotation curves of spiral galaxies) without adding unknown forms of dark energy or dark matter. In f(R) model, the Ricci scalar in the Einstein–Hilbert action is replaced by a general function of it. Its power-law form Rn is analyzed here using observed orbits of S-stars around the Galactic Center and also their computer simulations. We review the various consequences of the Rn gravity parameters (rc, the characteristic radius, i.e. the scalelength depending on the gravitating system properties; and β, the universal constant) on stellar dynamics and investigate their constraints from the observed S-star orbits. The presented results show that these observations could put reliable constraints on the parameters of Rn gravity.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2023

ECAR-7514 Rev 0 Analysis of the MARVEL Central Insurance Absorber Actuator Components

This ECAR analyzes the MARVEL Central Insurance Absorber (CIA) rod actuator design. Functions of interest include general vertical position, velocity, and SCRAM performance when carrying the CIA rod payload. Assessed components include the motor, drive train, electromagnet, bearings used for alignment and SCRAM velocity control, limit switches, 2nd position indicator, and resolver.

21 - SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLAN↗
DOE OSTI2018

Competing adsorption mechanisms of pyridine on Cu, Ag, Au, and Pt(110) surfaces

We explore the adsorption of pyridine on Cu, Ag, Au, and Pt(110) surfaces using density functional theory. To account for the van der Waals interaction, we use the optB86b-vdW, optB88-vdW, optPBE-vdW, revPBE-vdW, and rPW86-vdW2 functionals. For comparison, we also run calculations using the generalized gradient approximation-PBE (Perdew–Burke–Ernzerhof) functional. We find the most stable adsorption site to depend on both metal and functional, with two energetically favorable adsorption sites, namely, a vertically oriented site and a flat pyridine site. We calculate that every functional predicts pyridine to lie in the vertical configuration on the coinage metals at a low coverage. On Pt(110), by contrast, we calculate all the functionals—except rPW86-vdW2—to predict pyridine to lie flat at a low coverage. By analyzing these differences for these adsorption configurations, along with various geometric and electronic properties of the adsorbate/substrate system, we access in detail the performance of the 6 functionals we use. We also characterize the nature of the bonding of pyridine on the coinage metals from weak to strong physisorption, depending on the functional used. On Pt(110), we characterize the nature of the bonding of pyridine as ranging from strong physisorption to strong chemisorption depending again on the functional used, illustrating both the importance of the van der Waals interaction to this system and that this system can make a stringent test for computational methods.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Machine-learning-enabled on-the-fly analysis of RHEED patterns during thin film deposition by molecular beam epitaxy

Thin film deposition is a fundamental technology for the discovery, optimization, and manufacturing of functional materials. Deposition by molecular beam epitaxy (MBE) typically employs reflection high-energy electron diffraction (RHEED) as a real-time in situ probe of the growing film. However, the state-of-the-art for RHEED analysis during deposition requires human observation. Here, we present an approach using machine learning (ML) methods to monitor, analyze, and interpret RHEED images on-the-fly during thin film deposition. In the analysis workflow, RHEED pattern images are collected at one frame per second and featurized using a pretrained deep convolutional neural network. The feature vectors are then statistically analyzed to identify changepoints; these changepoints can be related to changes in the deposition mode from initial film nucleation to a transition regime, smooth film deposition, and in some cases, an additional transition to a rough, islanded deposition regime. The feature vectors are additionally analyzed via graph analysis and community classification. The graph is quantified as a stabilization plot, and we show that inflection points in the stabilization plot correspond to changes in the growth regime. The full RHEED analysis workflow is termed RHAAPsody and includes data transfer and output to a visual dashboard. We demonstrate the functionality of RHAAPsody by analyzing the precaptured RHEED images from epitaxial depositions of anatase TiO2 on SrTiO3(001) and show that the analysis workflow can be executed in less than 1 s. Our approach shows promise as one component of ML-enabled real-time feedback control of the MBE deposition process.

36 MATERIALS SCIENCE↗
DOE OSTI2017

Experimental and Modeling Studies for the Removal of Crystal Violet Dye from Aqueous Solutions using Eco-friendly Gracilaria corticata Seaweed Activated Carbon/Zn/Alginate Polymeric Composite Beads

This research article describes, an eco-friendly activated carbon prepared from the Gracilaria corticata seaweeds which was employed for the preparation of biodegradable polymeric beads for the efficient removal of crystal violet dye in an aqueous solution. The presence of chemical functional groups in the adsorbent material was detected using FTIR spectroscopy. The morphology and physical phases of the adsorbent materials were analyzed using SEM and XRD studies respectively. Batch mode dye adsorption behavior of the activated carbon/Zn/alginate polymeric beads was investigated as a function of dosage, solution pH, contact time, initial dye concentration and temperature. Maximum dye removal was observed at a pH of 5.0, 1 g of adsorbent dosage with 60 mg/L dye concentration, 50 min of contact time and at 30 °C. The equilibrium modeling studies were analyzed with Freundlich and Langmuir adsorption isotherms and the adsorption dynamics was predicted with Lagergren’s pseudo-first order, pseudo-second order equations and intra particle diffusion models. The process of dye removal followed a pseudo second-order kinetics rather than pseudo first order. The thermodynamic parameters like standard Gibbs free energy change (ΔG°), enthalpy change (ΔH°) and entropy change (ΔS°) were determined and the results imply that the adsorption process was spontaneous, endothermic and increases the randomness between the adsorbent and adsorbate. The results from the experimental and correlation data reveal that the Gracilaria corticata activated carbon/Zn/alginate polymeric beads have proved to be an excellent adsorbent material for the removal of CV dye.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Use Cases and Model Development of Thermal Storage Coupling for Advanced Nuclear Reactors

This report discusses the different options for coupling thermal energy storage (TES) systems to advanced nuclear power plants (A-NPPs) in order to enable flexible and hybrid plant operation. An advanced light-water reactor (ALWR) and a high-temperature gas-cooled reactor (HTGR) were selected as the initial use cases for demonstrating a thermally balanced energy storage coupling design for thermal power extraction. Cost functions for the A-LWR were derived from the fully balanced models that were developed based on three different coupling options with three different thermal energy bypass ratios. For the next steps, cost functions for the HTGR will also be derived, and additional nuclear reactors (e.g., a liquid-cooled fast reactor [LFR] or molten-salt reactor [MSR]) will be evaluated for coupling with TES in similar fashion, including the evaluation of their steady-state condition models and cost functions. The models presented herein showcase several design considerations, focusing on optimal deployment methodologies for achieving steady-state operation with minimum disruption to the nuclear power generation cycle. This report presents the results of steady state models developed using Aspen HYSYS®, wherein the thermal energy bypass for an NPP-TES coupling was varied up to 50%. The various components were sized using Aspen Process Economic Analyzer (APEA) and Aspen Exchanger Design and Rating (EDR), when applicable. Cost functions from these models were developed using the latest publicly available data obtained from APEA V11. The current steady-state models and cost functions provide a baseline for additional work focusing on dynamic operation and process optimization by using Idaho National Laboratory (INL)’s Framework for Optimization of Resources and Economics (FORCE) tools to evaluate the technoeconomic viability and transient operations of TES-coupled A-NPPs.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2018

Theoretical Studies of Elementary Hydrocarbon Species and Their Reactions

The research program supported by this DOE grant carried out both methodological development and computational applications of first-principles theoretical chemistry based on quantum mechanical wavefunctions, as directed toward understanding and harnessing the fundamental chemical physics of combustion. To build and refine the world’s database of thermochemistry, spectroscopy, and chemical kinetics, predictive and definitive computational methods are needed that push the envelope of modern electronic structure theory. The application of such methods has been made to gain comprehensive knowledge of the paradigmatic reaction networks by which the n- and i-propyl, t-butyl, and n-butyl radicals are oxidized by O 2 . Numerous ROO and QOOH intermediates in these R + O 2 reaction systems have been characterized along with the interconnecting isomerization transition states and the barriers leading to fragmentation. Other combustion-related intermediates have also been studied, including methylsulfinyl radical, cyclobutylidene, and radicals derived from acetaldehyde and vinyl alcohol. Theoretical advances have been achieved and made available to the scientific community by implementation into PSI4, an open-source electronic structure computer package emphasizing automation, advanced libraries, and interoperability. We have pursued the development of universal explicitly correlated methods applicable to general electronic wavefunctions, as well as a framework that allows multideterminant reference functions to be expressed as a single determinant from quasiparticle operators. Finally, a rigorous analytical tool for correlated wavefunctions has been created to elucidate dispersion interactions, which play essential roles in many areas of chemistry, but whose effects are often masked and enigmatic. Our research decomposes and analyzes the coupled-cluster electron correlation energy in molecular systems as a function of interelectronic distance. Concepts are emerging that can be used to explain the influence of dispersion on the thermochemistry of large hydrocarbons, including fuels important to combustion technologies.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Fast inversion, preconditioned quantum linear system solvers, fast Green's-function computation, and fast evaluation of matrix functions

Preconditioning is the most widely used and effective way for treating ill-conditioned linear systems in the context of classical iterative linear system solvers. We introduce a quantum primitive called fast inversion, which can be used as a preconditioner for solving quantum linear systems. The key idea of fast inversion is to directly block encode a matrix inverse through a quantum circuit implementing the inversion of eigenvalues via classical arithmetics. We demonstrate the application of preconditioned linear system solvers for computing single-particle Green's functions of quantum many-body systems, which are widely used in quantum physics, chemistry, and materials science. We analyze the complexities in three scenarios: the Hubbard model, the quantum many-body Hamiltonian in the plane-wave-dual basis, and the Schwinger model. We also provide a method for performing Green's function calculation in second quantization within a fixed-particle manifold and note that this approach may be valuable for simulation more broadly. Aside from solving linear systems, fast inversion also allows us to develop fast algorithms for computing matrix functions, such as the efficient preparation of Gibbs states. Furthermore, we introduce two efficient approaches for such a task, based on the contour-integral formulation and the inverse transform, respectively.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Extraction of Proton and Deuteron F2 Structure Function from Inclusive Electron-Nucleon Scattering at Large Bjorken-X

The E12-10-002 experiment measured the inclusive electron-nucleon cross section at Jeferson Lab at large Q2 (up to 16 GeV2) and a large range of x (up to 0.99). The data is analyzed and the cross sections are extracted for hydrogen and deuterium. Then the F2 structure function is extracted for hydrogen and deuterium. Finally the quark-hadron duality is tested for the proton F2 structure function for the previous world data and this experiment.

Biswas, Debaditya↗
DOE OSTI2022

First-principles calculation of gate-tunable ferromagnetism in magic-angle twisted bilayer graphene under pressure

Magic-angle twisted bilayer graphene (MATBG) is notable as a highly tunable platform for investigating strongly correlated phenomena such as unconventional superconductivity and quantum spin liquids, due to easy control of doping level through gating and sensitive dependence of the magic angle on hydrostatic pressure. Experimental observations of correlated insulating states, unconventional superconductivity and ferromagnetism in MATBG indicate that this system exhibits rich exotic phases. In this work, using density functional theory calculations in conjunction with the effective screening medium method, we find the MATBG under pressure at a twisting angle of 2.88 °and simulate how its electronic states evolve when doping level and electric field perpendicular to plane are tuned by gating. Our calculations show that, at doping levels between two electrons and four holes per moiré unit cell, a ferromagnetic (FM) solution with spin density localized at AA stacking sites is lower in energy than the nonmagnetic solution. The magnetic moment of this FM state decreases with both electron and hole doping and vanishes at four electrons/holes doped per moiré unit cell. Hybridization between the flat bands at the Fermi level and the surrounding dispersive bands can take place at finite doping. On increasing the out-of-plane electric field at zero doping, a transition from the FM state to the nonmagnetic one is seen. An investigation of impurity effects shows that both absorption of O 2 molecules and occurrence of Stone–Wales impurities suppress the FM state, and the mechanisms are understood from our calculations. We also analyze the interlayer bonding character due to flat bands via Wannier functions. Finally, we report trivial band topology of the flat bands in the FM state at a certain doping level.

Physics↗
DOE OSTI2021

Anion Exchange Ionomers: Impact of Chemistry on Thin-Film Properties

Ionomer thin-films (i.e., 20–100 nm) on supports serve as model systems to understand ionomer-catalyst interfacial behavior as well as the confinement-driven deviation in properties from bulk membranes. While ionomer thin-films have been examined for proton exchange ionomers, the thin-film properties of anion exchange ionomers (AEIs) remain largely unexplored. More importantly, delineating the convoluted impact of chemistry and confinement on thin-film morphology and hydration is of interest to advancing the field on functional ionic interfaces. In this work, these aspects are studied by using AEIs of different backbones (perfluorinated, aliphatic, and aromatic) and side chains (various lengths, and single versus dual functional groups). Quartz-crystal microbalance and spectroscopic ellipsometry are used to analyze density and coupled with calculated free volume fraction of thin-films to provide insights on their gas transport properties. AEI side-chain's chemical character plays a key role in how confinement modulates hydration (in thin-film versus bulk). Overall, the results elucidate the effects of backbone, side-chain chemistry versus anion/cation type in the confinement-driven changes in thin-film morphology and swelling. This study also provides new insights for tuning AEI transport functionalities at interfaces via chemistry, which can benefit the design and development of electrode-ionomers for alkaline membrane-based energy systems.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2024

How Precisely Can Individual Molecules Be Analyzed? A Case Study on Locally Quantifying Forces and Energies Using Scanning Probe Microscopy

Recent advances in scanning probe microscopy methodology have enabled the measurement of tip-sample interactions with picometer accuracy in all three spatial dimensions, thereby providing a detailed site-specific and distance-dependent picture of the related properties. This paper explores the degree of detail and accuracy that can be achieved in locally quantifying probe-molecule interaction forces and energies for adsorbed molecules. Toward this end, cobalt phthalocyanine (CoPc), a promising CO 2 reduction catalyst, was studied on Ag(111) as a model system using low-temperature, ultrahigh vacuum noncontact atomic force microscopy. Data were recorded as a function of distance from the surface, from which detailed three-dimensional maps of the molecule's interaction with the tip for normal and lateral forces as well as the tip-molecule interaction potential were constructed. The data were collected with a CO molecule at the tip apex, which enabled a detailed visualization of the atomic structure. Determination of the tip-substrate interaction as a function of distance allowed isolation of the molecule-tip interactions; when analyzing these in terms of a Lennard-Jones-type potential, the atomically resolved equilibrium interaction energies between the CO tethered to the tip and the CoPc molecule could be recovered. Interaction energies peaked at less than 160 meV, indicating a physisorption interaction. As expected, the interaction was weakest at the aromatic hydrogens around the periphery of the molecule and strongest surrounding the metal center. In conclusion, the interaction, however, did not peak directly above the Co atom but rather in pockets surrounding it.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Towards functionalized lignin and its derivatives for high-value material applications

In pursuit of low-carbon development, the production of diverse functional materials from renewable resources, especially lignocellulosic biomass, is of vital importance. Lignin is a major component of lignocellulose, and it is nature's only true high-volume aromatic polymer, which has the advantages of biodegradability, biocompatibility, and low acquisition cost. After modification by a variety of physicochemical strategies, it can be applied as an alternative to basic industrial materials, including polyurethanes, phenolic, and epoxy resins, which stands comparably to conventional synthetic polymers in terms of both performance and reduced cost. To further unlock the potential of lignin, an alluring opportunity is the exploitation of lignin for high-value materials, especially nanomaterials, that find extensive applications in energy and environment areas. This review provides a systematic summary and perspective of research that has been devoted to lignin transformation to high-value materials, ranging from industrially well-established engineering materials to the recently emerged nanomaterials that were used for energy and environment applications. Latest cutting-edge innovations on lignin modification, controlled depolymerization, and assembly during the last five years are summarized. Structure-function relationships of lignin materials in terms of their specific applications are analyzed. Furthermore, challenges and future opportunities for lignin conversion to high-value materials are also provided. We wish this review will stimulate further advances in lignin based high-value materials, and promote “waste” into “wealth”.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Study of Self-Interaction Errors in Density Functional Calculations of Magnetic Exchange Coupling Constants Using Three Self-Interaction Correction Methods

We examine the role of self-interaction errors (SIE) removal on the evaluation of magnetic exchange coupling constants. In particular we analyze the effect of scaling down the self-interaction-correction (SIC) for three non-empirical density functional approximations (DFAs) namely, the local spin density approximation, the Perdew-Burke-Ernzerhof generalized gradient approximation, and recent SCAN family of meta-GGA functionals. To this end, we employ three one-electron SIC methods: Perdew-Zunger [Perdew, J. P.; Zunger, A. Phys. Rev. B, 1981, 23, 5048] SIC, the orbitalwise scaled SIC method [Vydrov, O. A. et al., J. Chem. Phys. 2006, 124, 094108], and the recent local scaling method [Zope, R. R. et al., J. Chem. Phys. 2019, 151, 214108]. We compute the magnetic exchange coupling constants using the spin projection and non projection approaches for sets of molecules composed of dinuclear and polynuclear H-He models, organic radical molecules, and chlorocuprate, and compare these results against accurate theories and experiment. Our results show that for the systems that mainly consist of single electron regions, PZSIC performs well but for more complex organic systems and the chlorcuprates, an overcorrecting tendency of PZSIC combined with the DFAs utilized in this work is more pronounced, and in such cases LSIC with kinetic energy density ratio performs better than PZSIC. Furthermore, analysis of the results in terms of SIC corrections to the density and to the total energy shows that both density and energy correction are required to obtain an improved prediction of magnetic exchange couplings.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2026

Microstructure of amide-functionalized polyethylenes determined by NMR relaxometry

Amidation of polyethylenes creates a range of amide-containing materials with enhanced properties, but the effect of these functional groups on the microstructure of these new materials is not known. Here we employ solid-state nuclear magnetic resonance (NMR) techniques to analyze the microstructure of amide-modified polyethylenes. While a decrease in crystallinity was observed with increasing amounts of functionalization, we found by measuring the chain mobility of the crystalline, amorphous, and interphasial regions of the polyethylenes with NMR relaxation techniques that the grafted amidyl groups partition into the rigid amorphous fraction (RAF) between the crystalline and amorphous regions. The chemical specificity of these NMR experiments creates precise assessments of the location of functional groups within the materials. Together, these insights into the microstructure and morphology of amide-containing polyethylenes lay a foundation for a deeper understanding of the structure and properties of functional polyolefins.

Haber, Shira [Lawrence Berkeley National Laborator↗
DOE OSTI2022

Multilevel Analysis, Design, and Modeling of Coupling Advanced Nuclear Reactors and Thermal Energy Storage in an Integrated Energy System

This report discusses the different options for coupling thermal energy storage (TES) systems to advanced nuclear power plants (A-NPPs) in order to enable flexible and hybrid plant operation. An advanced light-water reactor (A LWR), a high-temperature gas-cooled reactor (HTGR) and a liquid-metal fast reactor (LMFR) were selected as the initial use cases for demonstrating a thermally balanced energy storage coupling design for thermal power extraction. The models presented herein showcase several design considerations, focusing on optimal deployment methodologies for achieving steady-state and transient-state operation with minimum disruption to the nuclear power cycle. This first part of the study presents steady-state models developed using Aspen HYSYS®, with the thermal energy bypass for NPP-TES coupling being varied at up to 50%. The various components were sized using the Aspen Process Economic Analyzer (APEA) and Aspen Exchanger Design and Rating (EDR), when applicable. Cost functions from these models were developed using the latest publicly available data obtained from APEA V11. The TES-coupled A-NPP steady-state models and cost functions then provided a baseline for dynamic operation and process optimization by using Idaho National Laboratory (INL)’s Framework for Optimization of Resources and Economics (FORCE) tools. A stochastic optimization of the various energy storage systems coupled to the A-NPPs was then performed using the Risk Analysis Virtual Environment (RAVEN) and its dispatch optimization plugin, the Holistic Energy Resource Optimization Network (HERON). The signal processing and synthetic history capabilities of RAVEN were used to account for the unpredictable behavior of electricity markets. An autoregressive moving average (ARMA) model was used to analyze price signals from the Pennsylvania-New Jersey-Maryland (PJM) market and were applied to the HERON analysis in order to optimize a system with the best economics. Transient modeling evaluation was then performed using Modelica models within the HYBRID repository, which was developed at INL for the Department of Energy Integrated Energy Systems program for the characterization of dynamic integrated system behavior and feedback. This includes evaluation of the TES-coupled A-LWR systems’ impact on physical and thermal system response during imposed system demands. Additional TES-coupled reactor types, coupling approaches, markets, and TES technologies will be evaluated in future work.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2024

Computationally efficient and error aware surrogate construction for numerical solutions of subsurface flow through porous media

Limiting the injection rate to restrict the pressure below a threshold at a critical location can be an important goal of simulations that model the subsurface pressure between injection and extraction wells. The pressure is approximated by the solution of Darcy’s partial differential equation for a given permeability field. The subsurface permeability is modeled as a random field since it is known only up to statistical properties. This induces uncertainty in the computed pressure. Solving the partial differential equation for an ensemble of random permeability simulations enables estimating a probability distribution for the pressure at the critical location. These simulations are computationally expensive, and practitioners often need rapid online guidance for real-time pressure management. An ensemble of numerical partial differential equation solutions is used to construct a Gaussian process regression model that can quickly predict the pressure at the critical location as a function of the extraction rate and permeability realization. The Gaussian process surrogate analyzes the ensemble of numerical pressure solutions at the critical location as noisy observations of the true pressure solution, enabling robust inference using the conditional Gaussian process distribution. Our first novel contribution is to identify a sampling methodology for the random environment and matching kernel technology for which fitting the Gaussian process regression model scales as O ( n log n ) instead of the typical O ( n 3 ) rate in the number of samples n used to fit the surrogate. The surrogate model allows almost instantaneous predictions for the pressure at the critical location as a function of the extraction rate and permeability realization. Our second contribution is a novel algorithm to calibrate the uncertainty in the surrogate model to the discrepancy between the true pressure solution of Darcy’s equation and the numerical solution. Finally, although our method is derived for building a surrogate for the solution of Darcy’s equation with a random permeability field, the framework broadly applies to solutions of other partial differential equations with random coefficients.

54 ENVIRONMENTAL SCIENCES↗